We first check that, for every \(\phi \in \operatorname{FMap} (\Sigma )\) , \({\phi }\)  is finite.
Suppose towards a contradiction that \({\phi }\)  is infinite. By replacing \(\phi \)  with \(\phi ^{-1}\) 
if necessary, we may assume that \(\phi \)  maps infinitely many points from \(E(z) \cap E_{-}\)  into
\(E_{+}\) . That is, there is a sequence \(z_i \in E(z) \cap E_{-}\)  such that \(\phi (z_i)\in E_{+}\) .
After taking a sub-sequence, we can assume \(z_i \rightarrow x \in X\) . Since \(\phi \)  is continuous,
and \(E_{+}\)  is closed, \(\phi (x) \in E_{+}\) . But this contradicts the fact that \(X \subset E_-\)  is \(\operatorname{FMap} (\Sigma )\) 
invariant.
The proof of the triangle inequality and the fact that \({\phi } = {\phi ^{-1}}\) 
are identical to the proof in Theorem REF .
Also, the subgroup of \(\operatorname{FMap} (\Sigma )\)  that fixes \(L\)  is a neighborhood of
the identity where \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)\( is zero, hence \)\(\centerdot \)
\(\centerdot \)
\( is continuous.\)
Proceeding as in Section , we now show that the length function \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)\( isbounded above by a uniform multiple of the word length.\)
Theorem 5.2 
For \(z \in E\) , assume \(E(z)\)  is two-sided and let \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)\( be the associated length function. Let \) S\( be a CB generating set for\)\(\operatorname{FMap}\)  ()\(, and let \)\(\centerdot \)
\(\centerdot \)
S\( denote the associated word length on \)\(\operatorname{FMap}\)  ()\(.Then there exists a constant \) c>0\( such that for every \)\(\operatorname{FMap}\)  ()\(\begin{equation}{\psi } \le c \cdot {\psi }_{\mathcal {S}}.\end{equation}\)
Since \(\operatorname{FMap} (\Sigma )\)  is \(CB\)  generated, the word length associated to every two CB generating sets are Lipschitz
equivalent. Hence, without loss of generality, we can assume \(\operatorname{FMap} (\Sigma )\)  is equipped with the generating set given
in Section REF . That is, there is a finite set \(F\)  such that \(\mathcal {S}= \nu _L \cup F\) .
For \(A \in \mathcal {A}\) , denote the complementary component of \((\Sigma - L)\)  associated to \(A\)  by \(\Sigma _A\) .
We further write
\(\nu _L = \cup _{A \in \mathcal {A}} \nu _A\) 
where elements of \(\nu _A\)  have support in \(\Sigma _A\) .
If \(s \in \nu _L\) , then \(s\)  fixes every \(A \in \mathcal {A}\)  set-wise. In particular, \(s\)  fixes \(E_+\)  and \(E_-\) 
set-wise. Therefore, \({s}=0\) . Now define
\(c = \max _{s \in F} {s}.\) 
Then, for \(s \in \mathcal {S}\) , \({s}\le c\) . For \(\phi \in \operatorname{FMap} (\Sigma )\) , if
\(\phi = s_1 \dots s_n\) ,
where \(s_i \in S\)  and \(n = {\phi }_{\mathcal {S}}\) , then
\({\phi } \le \sum _{i=1}^n {s_i} \le c \cdot n.\) 
That is, \({\phi } \le c \cdot {\phi }_{\mathcal {S}}\) .

Two-sided \(E^G\)
Assume \(E^G\)  is two-sided. Recall from the introduction that this implies that \(E^G = X \sqcup Y\) , where \(X, Y\)  are
non-empty disjoint closed \(\operatorname{FMap} (\Sigma )\) –invariant subsets of \(E\) . As in the previous section, for every
\(A \in \mathcal {A}\) , we have either \(X \cap A = \emptyset \)  or
\(Y\cap A =\emptyset \) . Therefore, we can choose a curve \(\beta \)  in \(L\)  giving a decomposition
\(\Sigma = \Sigma _- \cup \Sigma _+\) , where \(X \subset E_- = E(\Sigma _-)\)  and \(Y \subset E_+ = E(\Sigma _+)\) .
Heuristically, to follow the two-sided end case, for \(\phi \in \operatorname{FMap} (\Sigma )\)  we would like to count the genus of the
subsurface of \(\Sigma _-\)  that is moved by \(\phi \)  to \(\Sigma _+\)  plus the genus of the subsurface of \(\Sigma _+\) 
that is moved by \(\phi \)  to \(\Sigma _-\) . But this is not the correct measurement. For example, consider a large
genus subsurface \(Y\)  that intersects both \(\Sigma _-\)  and \(\Sigma _+\)  and let \(\phi \)  be a pseudo-Anosov
homeomorphism with support in \(Y\) . Then no subsurface of \(\Sigma _-\)  is moved to \(\Sigma _+\) .
Instead, \({\phi }\)  will be the genus of the subsurface \(Y\) .
We use the \(\mathbb {Z}_2\) –homology of the surface \(\Sigma \) , where \(\mathbb {Z}_2=\mathbb {Z}/2\mathbb {Z}\) .
We note that separating curves on infinite-type surfaces represent elements of homology which are not necessarily
zero (see [1]}). However, we quotient \(H_1(\Sigma , \mathbb {Z}_2)\)  by \(H_{1}^{sep}(\Sigma , \mathbb {Z}_2)\) , the subgroup
of \(H_1(\Sigma , \mathbb {Z}_2)\)  generated by homology classes that can be represented
by simple separating closed curves on the surface. We denote this quotient by
\(\widehat{H} = \widehat{H}_1(\Sigma , \mathbb {Z}_2) = H_1(\Sigma , \mathbb {Z}_2)/H_{1}^{sep}(\Sigma , \mathbb {Z}_2).\) 
Since the curve \(\beta \)  represents a trivial class in \(\widehat{H}\) , the decomposition \(\Sigma = \Sigma _+ \cup \Sigma _-\) 
gives a splitting \(\widehat{H} = H_+ \oplus H_-\)  where
\(H_{+} = \widehat{H}_1(\Sigma _{+}, \mathbb {Z}/2)\qquad \text{and}\qquad H_{-} = \widehat{H}_1(\Sigma _{-}, \mathbb {Z}/2).\) 
Note that \(H_- \cap H_+ = \emptyset \)  since \(\Sigma _-\)  and \(\Sigma _+\)  are disjoint.
For \(\phi \in \operatorname{FMap} (\Sigma )\) , we denote the induced map on homology by
\(\phi ^{*} \colon \,H_1(\Sigma , \mathbb {Z}/2) \rightarrow H_1(\Sigma , \mathbb {Z}/2)\) . Since \(\phi \)  sends separating
curves to separating curves, we also have an induced isomorphism \(\phi ^* \colon \,\widehat{H} \rightarrow \widehat{H}\) .
Define \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)   \(\operatorname{FMap}\)  () Z+\( by\begin{equation} {\phi } = \operatorname{codim}\Big ( \widehat{H} : \big (H_+ \cap \phi ^*(H_+)\big ) \oplus \big (H_- \cap \phi ^*(H_-)\big ) \Big ),\end{equation}where \) codim(V : W)\( is the co-dimension of a subspace \) W\( in a vector space \) V\(.\)
Theorem 5.3 
Assume \(E^G\)  is two-sided. Then \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)   \(\operatorname{FMap}\)  () N\( is a length function.\)
We start by showing that, for \(\phi \in \operatorname{FMap} (\Sigma )\) , \({\phi }\)  is finite.
Since \(\phi \)  fixes \(X\)  set-wise, there is a small neighborhood of \(X\)  in \(\Sigma _-\)  that is contained in
\(\Sigma _-\) . That is, there is a subsurface \(\Sigma ^{\prime } \subset \Sigma _-\)  such that \(X \subset E(\Sigma ^{\prime })\)  and
\(\phi (\Sigma ^{\prime }) \subset \Sigma _-\) . The subsurfaces \((\Sigma _- - \Sigma ^{\prime })\)  and \((\Sigma _- - \phi (\Sigma ^{\prime }))\)  both have
finite genus because the fact that \(E^G \cap E(\Sigma _-) = X\) 
implies that their end space is disjoint from \(E^G\) 
This means that
\(\operatorname{codim}(H_-: \widehat{H}_1(\Sigma ^{\prime }, \mathbb {Z}_2)) < \infty \qquad \text{and}\qquad \operatorname{codim}(H_-: \widehat{H}_1(\phi (\Sigma ^{\prime }), \mathbb {Z}_2) ) < \infty .\) 
But
\(\widehat{H}_1(\Sigma ^{\prime }, \mathbb {Z}_2) \cap \widehat{H}_1(\phi (\Sigma ^{\prime }), \mathbb {Z}_2) \subset H_- \cap \phi ^*(H_-).\) 
Therefore,
\(\operatorname{codim}(H_-: H_- \cap \phi ^*(H_-)) \le \operatorname{codim}(H_-: \widehat{H}_1(\Sigma ^{\prime }, \mathbb {Z}_2)) +\operatorname{codim}(H_-: \widehat{H}_1(\phi (\Sigma ^{\prime }), \mathbb {Z}_2) ) < \infty .\) 
Similarly, \(\operatorname{codim}(H_+: H_+ \cap \phi ^*(H_+)) < \infty \)  and hence \({\phi } < \infty \) .
We now check that \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)\( satisfies the triangle inequality. Consider \) , \(\operatorname{FMap}\)  ()\(.We have{\begin{@align*}{1}{-1}\operatorname{codim}\big (H_\pm &: H_\pm \cap \psi (\phi (H_\pm )) \big )\le \operatorname{codim}\big (H_\pm : H_\pm \cap \psi (H_\pm ) \cap \psi (\phi (H_\pm )) \big ) \\&= \operatorname{codim}\big (H_\pm : H_\pm \cap \psi (H_\pm ) \big )+ \operatorname{codim}\big (H_\pm \cap \psi (H_\pm ) : H_\pm \cap \psi (H_\pm ) \cap \psi (\phi (H_\pm )) \big )\\&\le \operatorname{codim}\big (H_\pm : H_\pm \cap \psi (H_\pm ) \big )+ \operatorname{codim}\big (\psi (H_\pm ) : \psi (H_\pm ) \cap \psi (\phi (H_\pm )) \big )\\&= \operatorname{codim}\big (H_\pm : H_\pm \cap \phi (H_\pm ) \big )+ \operatorname{codim}\big (H_\pm : H_\pm \cap \phi (H_\pm ) \big ).\end{@align*}}Summing over \)\( we get\)\({\psi \circ \phi } \le {\psi } + {\phi }.\)\(\) The subgroup \(\nu _L\)  of \(\operatorname{FMap} (\Sigma )\)  is a neighborhood of the identity and for \(\phi \in \nu _L\) ,
\(\phi \)  preserves \(\Sigma _\pm \) . Hence, \(\phi ^*\)  fixes \(H_\pm \)  and \({\phi }=0\) .
This fact and the triangle inequality imply that \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)\( is continuous. Also, since\)\((\phi ^*)^{-1} (H_\pm \cap \phi ^*(H_\pm )) = (\phi ^*)^{-1} (H_\pm ) \cap (H_\pm )\)\(we have \)  = -1\(. Therefore \)\(\centerdot \)
\(\centerdot \)
\(is a length function.\)
We now show that the function \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)\( is bounded above by the word metric on \)\(\operatorname{FMap}\)  ()\(.\)
Theorem 5.4 
There exist a constant \(c >0\)  such that
\({\phi } \le c \cdot {\phi }_{\mathcal {S}},\) 
where \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\) S\( denotes the word metric on \)\(\operatorname{FMap}\)  ()\(.\)
This proof follows the same reasoning as the proof of Theorem REF ,
except we use that \(E^G\)  is 2-sided in place of \(E(z)\)  being two-sided.
As before, we assume that \(\operatorname{FMap} (\Sigma )\)  is equipped with the generating set \((\nu _L, F)\)  given in Section
REF . If \(s\in \nu _L\) , then \(s\)  fixes
the curve \(\beta \)  which is contained in \(L\)  and hence it fixes \(\Sigma _\pm \) .
In particular, \(s^*\)  fixes \(H_\pm \) . Therefore, \({\phi }=0\) . Now, for
\(c = \max _{\phi \in F} {\phi },\) 
we have \({\phi } \le c \cdot {\phi }_{\mathcal {S}}\) .

Infinite asymptotic dimension
In this section, we prove that the mapping class group of an infinite type surface \(\Sigma \)  with a two-sided end space
has infinite asymptotic dimension. For any such surface, we construct an associated shift map and
we use the length function from Section  to show that they are essential. We then follow
the arguments in Section  to embed \({Q^{\infty }}\)  in \(\operatorname{FMap} (\Sigma )\) .
Proposition 6.1 
If \(E=E(\Sigma )\)  is two-sided, then \(\operatorname{FMap} (\Sigma )\)  contains an essential shift.

Assume first that \(E^G\)  is two-sided. That is, \(E^G = X \sqcup Y\)  where \(X\)  and \(Y\)  are non-empty closed
\(\operatorname{FMap} (\Sigma )\)  invariant sets. Let \(\beta \)  be a curve in \(L\)  separating \(X\)  from \(Y\)  and, as before,
let \(\Sigma _-\)  and \(\Sigma _+\)  be the components of \(\Sigma -\beta \) . Also, let \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)\( be the normdefined in Theorem~\ref {Thm:E^G-length-function}.\) Pick \(x \in X\)  and \(y \in Y\) . Since every neighborhood of \(x\)  and \(y\)  in \(\Sigma \) 
have non-zero genus, we can find a sequence of disjoint surfaces \(\Sigma _i\) , each homeomorphic to a genus
one surface with one boundary component, such that \(\Sigma _i \rightarrow x\)  as \(i \rightarrow -\infty \)  and \(\Sigma _i \rightarrow y\) 
as \(i \rightarrow \infty \) . We choose these so that \(\Sigma _i \in \Sigma _-\)  for \(i \le 0\)  and \(\Sigma _i \in \Sigma _+\) 
for \(i > 0\) .
We connect \(\Sigma _{i-1}\)  to \(\Sigma _i\)  by an arc \(\omega _i\)  so that the arcs \(\omega _i\)  are
disjoint from each other and the other \(\Sigma _j\) , and so only \(\omega _0\)  intersects \(\beta \) .
Let \(\sigma \)  be a regular neighborhood of the union of the \(\Sigma _i\)  and \(\omega _i\) . Then \(\sigma \)  is a strip
of infinite genus exiting towards \(x\)  and \(y\)  in each end and intersecting \(\beta \)  in an arc \(\beta |_\sigma \) .
Denote the components of \(\sigma - \beta |_\sigma \)  by \(\sigma _-\)  and \(\sigma _+\) .
Let \(h_\sigma \)  be the shift map with support in \(\sigma \)  sending \(\Sigma _i\)  to \(\Sigma _{i+1}\) .
Choosing a basis for the homology of each \(\Sigma _i\)  and extending it to the homology of \(\Sigma _-\) 
and \(\Sigma _+\) , we can decompose the homology of \(\Sigma _\pm \)  as follow:
\(H_- = \overline{H}_- \oplus \bigoplus _{i=-\infty }^{0} H_i\qquad \text{and}\qquad H_+ = \overline{H}_+ \oplus \bigoplus _{i=1}^{\infty } H_i\) 
where \(H_i = \widehat{H}(\Sigma _i, \mathbb {Z}_2)\)  and \(\overline{H}_\pm \)  are the homology of \(\Sigma _\pm -\sigma \) .
Then, \(h_\sigma ^*\)  fixes \(\overline{H}_\pm \)  and sends \(H_i\)  to \(H_{i+1}\) . Therefore,
for \(n>0\)
\((h_\sigma ^*)^n(H_-) = H_- \oplus \bigoplus _{i=1}^{n} H_i.\qquad \text{and}\qquad (h_\sigma ^*)^n(H_+) = \overline{H}_+ \oplus \bigoplus _{i=n+1}^{\infty } H_i\) 
That is,
\({h_\sigma ^n} =\operatorname{codim}\Big ( \widehat{H} : \big (H_+ \cap (h_{\sigma }^*)^n(H_+)\big ) \oplus \big (H_- \cap (h_{\sigma }^*)^n(H_-)\big ) \Big )= \dim \left( {\bigoplus _{i=1}^{n} H_i}\right) =2n.\) 
Theorem REF  implies that the diameter of the group \(\langle h_\sigma \rangle \)  is infinite.
Hence, by definition, \(\langle h_\sigma \rangle \)  is not CB and hence \(h_\sigma \)  is essential.
Now assume \(E(z)\)  is two-sided. That is, \(\mathrm {Accu} (z)= X \sqcup Y\)  where \(X\)  and \(Y\)  are non-empty closed
\(\operatorname{FMap} (\Sigma )\)  invariant sets. The proof in this case is similar. Let \(\beta \)  be a curve in \(L\)  separating \(X\)  from
\(Y\) , let \(\Sigma _-\)  and \(\Sigma _+\)  be the components of \(\Sigma -\beta \) , and let \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)\( be the normdefined in Theorem~\ref {Thm:z-length-function}. Pick \) x X\( and \) y Y\(. Since every neighborhood of \) x\( and \) y\(in \)\( has a point in \) E(z)\(, we can find a sequence of points \) zi E(z)\( where \) zi x\(as \) i -\( and \) zi y\( as \) i \(, where \) zi E-\( for \) i 0\( and \) zi +\(for \) i > 0\(. Choose disjoint stable neighborhoods \) Ei\( of \) zi\( so that \) M(Ei) = zi\(; this is possiblesince \) E(z)\( is countable. Let \) i\( be a subsurface of \)\( with one boundary component and with \) E(i)=Ei\(. We can assume the \) i\( are disjoint from each other and from \)\(.Then \) i x\( as \) i -\(, \) i y\( as \) i \(, \) i -\( for \) i < 0\(,and \) i +\( for \) i 0\(.\) As before, connect \(\Sigma _{i-1}\)  to \(\Sigma _i\)  by an arc \(\omega _i\)  so that the \(\omega _i\)  are
disjoint from each other and the other \(\Sigma _j\) , and so only \(\omega _0\)  intersects \(\beta \) .
Let \(\sigma \)  be a regular neighborhood of the union of \(\Sigma _i\)  and \(\omega _i\) . Then \(\sigma \)  is a strip
containing \(\Sigma _i\)  exiting towards \(x\)  and \(y\)  in each end and intersecting \(\beta \)  in an arc \(\beta |_\sigma \) .
Then, \(h_\sigma \)  acts as a shift on \(z_i\)  with \(h_\sigma ^n(z_i)=z_{i+n}\) . Therefore,
\({h_\sigma ^n}=n,\) 
which implies \(\langle h_\sigma \rangle \)  has infinite diameter and \(h_\sigma \)  is essential.
Similar to the construction of a puncture permutation map from Section , we now define a
subsurface permutation map for the strip \(\sigma \)  constructed above
where we replace the punctures with subsurfaces \(\Sigma _i\) .
Again, we make use of the function \(z \colon \,{Q^{\infty }}\rightarrow \mathbb {Z}\)  defined in Equation (REF ) which, for \(a \in {Q^{\infty }}\) , gives
the number of zeros in \(a\)  before the final one and the function \(z_{a}\colon \,\lbrace 1,...,z(a) \rbrace  \rightarrow \mathbb {N}\) 
defined in the equation Equation (REF ) which gives the positions of the zeros in \(a\) . The homeomorphism \(\pi _a\)  is a
composition of twists on the infinite strip \(\sigma \) . We define a \(1/n^{\text{th}}\) -twist around consecutive subsurfaces
\(\Sigma _{i+1}, \ldots , \Sigma _{i+n}\) , denoted by \(T^{1/n}_{i+1, \ldots , i+n}\) , as follows. Let \(c\)  be a curve surrounding
the \(n\)  subsurfaces as shown in Figure REF . Then, the twist map is the homeomorphism
with support on the surface separated by \(c\)  that sends \(\Sigma _j\)  to \(\Sigma _{j+1}\) 
for \(j=i+1, \dots , (n+i-1)\)  and sends \(\Sigma _{i+n}\)  to \(\Sigma _{i+1}\)  such
that \((T^{1/n}_{{i+1}, \ldots , {i+n}})^n\)  is a Dehn twist around the curve \(c\) .
We then define
\(\pi _a = \prod _{i=1}^{z(a)}T_{{z_a(i)}, \ldots , {{a}+i}}^{1/({a} + i - z_a(i))}.\) 
<FIGURE>Observe that, the \(i\) –th twist in the the above product moves the
\(\Sigma _{|a|+i}\)  to \(\Sigma _{z_a(i)}\) . Therefore, the surfaces
\(\Sigma _{{a}+1}, \dots , \Sigma _{{a}+ z(a)}\)  are sent by \(\pi _a\) 
to surfaces in correspondence with the positions where \(a\)  has zeros.
We finally define:
\(\begin{aligned}\Phi \colon \,{Q^{\infty }}&\rightarrow \operatorname{FMap} (\Sigma )\\a &\mapsto \pi _{a} h_{\sigma }^{{a}}.\end{aligned}\) 
<FIGURE>Using the same reasoning as in Section , we can see that
\(\Phi (a)\)  sends some subsurface \(\Sigma _j\)  from \(\Sigma _-\)  to
\(\Sigma _+\)  for every index \(i>0\)  where \(a_i=1\)  and sends this subsurface to \(\Sigma _i\) . If \(b_i=1\)  then
\(\Sigma _i\)  is sent back to \(\Sigma _-\)  under \(\Phi (b)^{-1}\) , otherwise
it stays in \(\Sigma _+\) . If \(a_i=0\)  the suburface sent to \(\Sigma _i\) 
by \(\Phi (a)\)  comes from \(\Sigma _+\) . Now if \(b_i=1\) , then \(\Sigma _i\) 
is sent to \(\Sigma _-\)  under \(\Phi (b)^{-1}\) . That is, the number
of surfaces that are sent from \(\Sigma _\pm \)  to \(\Sigma _\mp \) 
under \(\Phi (b)^{-1}\Phi (a)\)  is exactly the number of indices \(i\) 
where \(a\)  and \(b\)  are different, which is \(|a-b|\) .
When \(E^G\)  is two sided, the surfaces \(\Sigma _i\) 
are genus one surfaces with one boundary component. Therefore,
sending one of these surfaces from \(\Sigma _\pm \)  to \(\Sigma _\mp \)  contributes
2 to the norm defined in Equation () and we have
\({\Phi (b)^{-1}\Phi (a)(\beta )} = 2|a-b|.\) 
In the case where \(E(z)\)  is two-sided, \(\Sigma _i\)  has one maximal end which
is a point in \(E(z)\) . Therefore,
sending one of these surfaces from \(\Sigma _\pm \)  to \(\Sigma _\mp \)  contributes
1 to the norm defined in Equation () and we have
\({\Phi (b)^{-1}\Phi (a)(\beta )} = |a-b|.\) 
We now show that \(\Phi \)  is a quasi-isometric embedding from \({Q^{\infty }}\)  and
\(\operatorname{FMap} (\Sigma )\) .
Proposition 6.2 
Assume \(E = E(\Sigma )\)  is two-sided and let \(\Phi \colon \,{Q^{\infty }}\rightarrow \operatorname{FMap} ( \Sigma )\) 
be the map from Equation REF . Then there exists \(C>0\) 
such that, for every \(a,b \in {Q^{\infty }}\)  we have:
\(\frac{1}{C} \cdot {a-b}\le {\Phi (a) \Phi (b)^{-1}}_{\mathcal {S}} \le C \cdot {a-b}+3\) 
and hence \(\Phi \)  is a quasi-isometric embedding from
\((Q^{\infty }, {{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\) )\( to \) (\(\operatorname{FMap}\)  (), \(\centerdot \)
\(\centerdot \)
S)\(.\)
To obtain inequality on the left, we assume \(C\)  is larger
than the constants in Theorems REF 
and REF . We then combine
Equation (REF ) with
Theorem REF 
in the case where \(E^G\)  is two-sided
and we combine Equation (REF )
with Theorem REF 
in the case where \(E(z)\)  is
two-sided.
Recall that \(x,y \in E\)  are the ends of \(\Sigma \)  associated to \(\sigma \) .
Let \(A \in \mathcal {A}\)  be the set containing \(x\)  and \(B \in \mathcal {A}\)  be the
set containing \(y\) . We now argue as in the proof of Theorem REF , but in place
punctures, we use disjoint subsurfaces \(\Sigma _i\) . We find homeomorphisms
\(s_1, s_3 \in \nu _B\)  and \(s_2 \in \nu _A\)  such that
\((s_3 h_\sigma ^{k_2} s_2 h_\sigma ^{k_1} s_1) \circ \Phi (a) \Phi (b)^{-1} = id,\) 
where \(k_1\)  is the number indices \(i\)  where \(a_i=1\)  and \(b_i =0\)  and
\(k_2\)  is the number of indices where \(a_i=0\)  and \(b_i=1\) .
Further assuming that \(C \ge {h_\sigma }_\mathcal {S}\) , we have
\({\Phi (a) \Phi (b)^{-1}}_\mathcal {S}\le C \cdot (k_1 + k_2) + 3= C \cdot |a-b| + 3\) 
since the \(s_i\)  are generators.
This finishes the proof.
Theorem REF , restated below, now follows immediately.
Theorem 6.3 
Assume \(\Sigma \)  has a two-sided end space. Then
\(\operatorname{FMap} (\Sigma )\)  and \(\operatorname{Map} (\Sigma )\)  have infinite asymptotic dimension.

By Proposition REF , the map \(\Phi \colon \,({Q^{\infty }}, {{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\) ) (\(\operatorname{FMap}\)  ( ), dS)\(is a quasi-isometric embedding and \) Q\( has infinite asymptotic dimension byTheorem~\ref {Thm:Infinite-Asymptotic-Dimension-Qinf}. Therefore \)\(\operatorname{FMap}\)  ()\( also has infinite asymptotic dimension.\) By Theorem REF , \(\operatorname{FMap} (\Sigma )\)  and
\(\operatorname{Map} (\Sigma )\)  are quasi-isometric. Fact \(2.2\)  tells us that
asymptotic dimension is preserved under quasi-isometries, and hence
\(\operatorname{Map} (\Sigma )\)  has infinite asymptotic dimension.

Equivalence of Algebraically and Topologically Essential
In this section we prove Theorems REF  and REF 
from the introduction. Let \(h_\sigma \)  be a shift map as described in the introduction. Recall that the support
of \(h_\sigma \)  is a strip \(\sigma \) , containing a collection of sub-surfaces \(\Sigma _i\) , exiting towards points
\(x,y \in E\) .
We make use of the following theorem of Rosendal giving an equivalent condition for the notion of
coarsely bounded sets that is more suitable for our purposes.
Theorem 7.1 (Rosendal, Prop. 2.7 (5) in [2]})  
Let \(A\)  be a subset of a Polish group \(G\) . The following are equivalent

\(A\)  is coarsely bounded.

For every neighborhood \(\nu \)  of the identity in \(G\) , there is a finite subset \(\mathcal {F}\) 
and some \(k \ge 1\)  such that \(A \subset (\mathcal {F}\nu )^k\) .


We prove several special cases of Theorem REF . We then combine them to give a proof of the general case.
Proposition 7.2 
Assume \(\Sigma _i\)  is a surface of genus \(g\)  with one boundary component.
If \(h_\sigma \)  is essential, then \(E^G\)  is two-sided giving a decomposition \(E^G = X \sqcup Y\) 
such that \(x \in X\)  and \(y \in Y\) . In particular, if \(E^G\)  is not two-sided, then \(h_\sigma \) 
is not essential.

Let \(A, B \in \mathcal {A}\)  be such that \(x \in A\)  and \(y \in B\) . There is a homeomorphic copy \(\sigma ^{\prime }\) 
of \(\sigma \)  which is the concatenation of an infinite genus half-strip in \(\Sigma _A\) ,
a zero genus compact strip in \(L\) , and an infinite genus half-strip in \(\Sigma _B\) .
Let \(g\)  be a homeomorphism sending \(\sigma \)  to \(\sigma ^{\prime }\) . For \(\mathcal {F}\)  and \(\nu \)  as in
Theorem REF , let \(\mathcal {F}^{\prime } = g^{-1} \mathcal {F}g\)  and \(\nu ^{\prime } = g^{-1} \nu g\) . Then
\(h_\sigma ^n \in (\mathcal {F}\nu )^k\qquad \Longleftrightarrow \qquad h_{\sigma ^{\prime }}^n \in (\mathcal {F}^{\prime } \nu ^{\prime })^k.\) 
That is, \(\langle h_\sigma \rangle \)  is CB if and only if \(\langle h_{\sigma ^{\prime }} \rangle \)  is CB.
In fact, in general, any set conjugate to a CB set is CB. Hence, it is enough to prove the Proposition for \(h_{\sigma ^{\prime }}\) .
Claim:
Assume there is no decomposition \(E^G = X \sqcup Y\)  where
\(x\in X\) , \(y \in Y\)  and \(X,Y\)  are closed \(\operatorname{FMap} (\Sigma )\) –invariant sets.
Then, there is a sequence
\(A = A_0, A_1, \dots , A_k = B, \qquad A_i \in \mathcal {A}\) 
and a sequence of ends \(z_i \in E^G\)  and \(z_i^{\prime } \in E(z_i)\) , \(i = 0, \dots , (k-1)\)  such that
\(z_0 \in A_0-\mathcal {M}(A_0)\) , \(z_{k-1}^{\prime } \in A_k-\mathcal {M}(A_k)\)  and, for \(i = 1, \dots , (k-1)\)  ,
\(z_{i-1}^{\prime }, z_i \in A_i-\mathcal {M}(A_{i-1})\) .
To see that this claim holds, let \(\mathcal {A}^{\prime }\)  be the subset of \(\mathcal {A}\)  such that, for \(C \in \mathcal {A}^{\prime }\) , there is a sequence
as described above starting from \(A\)  and ending in \(C\) . Let \(\mathcal {B}^{\prime } = \mathcal {A}- \mathcal {A}^{\prime }\) .
Then, for every \(A^{\prime } \in \mathcal {A}^{\prime }\) , \(B^{\prime } \in \mathcal {B}^{\prime }\)  and \(z \in E_{A^{\prime }}\cap E^G\)  we have
\(E(z) \cap E_{B^{\prime }} = \emptyset \) . Define
\(X = \bigcup _{A^{\prime } \in \mathcal {A}^{\prime }} E_{A^{\prime }} \cap E^G\qquad \text{and}\qquad Y = \bigcup _{B^{\prime } \in \mathcal {B}^{\prime } } E_{B^{\prime }}\cap E^G.\) 
The sets \(X\)  and \(Y\)  are closed and \(\operatorname{FMap} (\Sigma )\)  invariant, and the definition of \(X\)  implies that
we have \(x \in X\) . The assumption implies that \(y\)  cannot be in \(Y\) . Therefore \(B \in \mathcal {A}^{\prime }\)  and the
above sequence exists, which proves the claim.
<FIGURE>For \(i=0, \dots , (k-1)\) , fix disjoint sub-surfaces \(Z_i \subset \Sigma _{A_i} -x_{A_i}\)  and
\(Z_i^{\prime }\subset \Sigma _{A_{i+1}}-x_{A_{i+1}}\)  such that \(z_i\)  is an end of \(Z_i\) , \(z_i^{\prime }\)  is an end of \(Z_i^{\prime }\)  and
\(Z_i\)  is homeomorphic to \(Z_i^{\prime }\) , for example, we can choose \(Z_i\)  so that \(E(Z_i)\)  is an stable neighborhood
of \(z_i\) . Connect \(Z_i\)  to \(Z_i^{\prime }\)  by an arc \(\omega _i\)  that is contained in
\(L \cup \Sigma _{A_i} \cup \Sigma _{A_{i+1}}\) . Let \(Y_i\)  be a regular neighborhood of
\(Z_i \cup Z_i^{\prime } \cup \omega _i\) . Further, we assume that subsurfaces \(Y_i\)  are disjoint from each other
and are disjoint from the strip \(\sigma ^{\prime }\)  (see Figure REF ). For \(i =1, \dots (k-1)\) , let \(\partial _i\) 
be a sub-arc of \(\partial \Sigma _{A_i}\)  connecting \(\omega _{i-1} \cap \partial \Sigma _{A_i}\)  to
\(\omega _{i} \cap \partial \Sigma _{A_i}\) . Also, let \(\omega _i|_L\)  be the restriction
of \(\omega _i\)  to \(L\) . We further assume that the concatenation of \(\omega _i|_L\)  and \(\partial _i\) ,
which is an arc connecting \(\partial \Sigma _A\)  to \(\partial \Sigma _B\) , is homotopic (relative
\(\partial \Sigma _A \cup \partial \Sigma _B\) ) to either arc in the the restriction of \(\partial \sigma ^{\prime }\)  to
\(L\) , which we denote by \(\partial \sigma ^{\prime }|_L\) . Let \(\partial _0\)  be the sub-arc of \(\partial \Sigma _A\) 
connecting a point in \(\partial \sigma ^{\prime } \cap \partial \Sigma _A\)  to the points \(\omega _0 \cap \partial \Sigma _A\) 
and similarly let \(\partial _k\)  be the sub-arc of \(\partial \Sigma _B\)  connecting a point in
\(\partial \sigma ^{\prime } \cap \partial \Sigma _B\)  to the points \(\omega _{k-1} \cap \partial \Sigma _B\) .
Again, these can be chosen such that the concatenation \((\partial _i \omega _i|_L )_{i=0}^{(k-1)}\) 
is homotopic to the concatenation of an arc in \(\partial \sigma ^{\prime }|_L\)  and \(\partial _k\)  relative their end points.
Let \(g_i\)  be a homeomorphism that sends \(Z_i\)  to \(Z_i^{\prime }\) , whose support
is in \(Y_i\)  and that sends \(\omega _i\)  to itself in the reverse direction.
Let \(\mathcal {F}= \lbrace g_i\rbrace _{i=0}^{(k-1)}\) .
We now show that, for every \(n \in \mathbb {Z}\) , \(h_{\sigma ^{\prime }}^n \in (\nu _L \mathcal {F})^{k+1}\) . Let \(\sigma ^{\prime }_n\)  be
a sub-strip of \(\sigma ^{\prime }\)  of genus \(n\)  so that the genus of what remains between \(\sigma _n^{\prime }\)  and
\(\partial A\)  is zero. Choose an element \(f_0 \in \nu _{A_0}\)  that sends \(\sigma ^{\prime }_n\)  to a
\(Z_0\) , which is possible since \(z_0 \in E^G\)  and therefore \(Z_0\)  has infinite genus. In fact, we can do this in a way such that
\(f_0(\sigma ^{\prime })\cap \Sigma _A\)  starts the same as \(\sigma ^{\prime }\cap \Sigma _A\)  away from a small neighborhood of
\(\partial \Sigma _A\) , then it follows \(\partial _0\)  (staying in a small neighborhood of \(\partial \Sigma _A\) )
to the intersection point of \(\partial \Sigma _A\)  and \(\omega _0 \cap \Sigma _A\) , then follows
\(\omega _0 \cap \Sigma _A\)  (staying in \(Y_0\) ) to \(Z_0\) , then comes back the same way along
\(\omega _0 \cap \Sigma _A\)  and \(\partial _0\)  and continues along \(\sigma ^{\prime }\)  into \(B\) 
(see the left picture in Figure REF ). Then, \(g_0 \circ f_0(\sigma _n^{\prime }) \subset Z_0^{\prime }\) .
<FIGURE>We then find a \(f_1 \in \nu _{A_1}\)  that sends \(g_0 \circ f_0(\sigma _n^{\prime })\)  to \(Z_1\) . We can do this in a way such that
\(f_1 \circ g_0 \circ f_0 (\sigma ^{\prime })\)  starts the same as \(\sigma ^{\prime }\)  inside \(\Sigma _A\)  (away from a small neighborhood
of \(\partial \Sigma _A\) ) then it follows \(\partial _0\) , then \(\omega _0|_L\) , then \(\partial _1\)  to the intersection
point of \(\partial \Sigma _{A_1}\)  and \(\omega _1\) , then it follows \(\omega _1 \cap \Sigma _{A_1}\) 
(staying in \(Y_1\) ) to \(Z_1\) , then comes back the same way along \(\omega _1 \cap \Sigma _{A_1}\) , \(\partial _1\) ,
\(\omega _0|_L\)  and \(\partial _0\) , and then follows \(\sigma ^{\prime }\)  into \(B\) . We then apply \(g_1\)  and we have
\(g_1 \circ f_1 \circ g_0 \circ f_0(\sigma _n^{\prime }) \subset Z_1^{\prime }\) .
Following the same argument, we can find \(f_i\) , \(i =0, \dots , (k-1)\)  such that
\(\left( \prod _{i=0}^{k-1} g_i \circ f_i\right) (\sigma _n^{\prime }) \subset Z_{k-1}^{\prime } \subset \Sigma _{A_k} = \Sigma _B.\) 
The strip \(\prod g_i \circ f_i(\sigma ^{\prime })\)  starts the same as \(\sigma ^{\prime }\)  in \(\Sigma _A\) , then
follows the concatenation of the arcs \(\omega _i|_L\)  and \(\partial _i\)  (which by assumption is homotopic to arcs
\(\partial \sigma ^{\prime }|_L\) ), then follows \(\omega _k\cap \Sigma _B\)  to \(Z_k\) , then back the same way to
\(\partial \Sigma _A \cap \sigma ^{\prime }\) , and then continues the same as \(\sigma ^{\prime }\)  (see the right picture in
Figure REF ).
The portion of \(\prod _{i=0}^{k-1} g_i \circ f_i (\sigma _n^{\prime })\)  traveling back from \(\partial \Sigma _B\) 
to \(\partial \Sigma _A\)  has genus zero and can be homotoped into \(\Sigma _B\) . The portion going forward from
\(\partial \Sigma _A\)  to \(\partial \Sigma _B\)  also has genus zero and can be homotoped to a neighborhood of
the concatenation of \(\partial \sigma ^{\prime }|_L\)  and \(\partial _k\) . Hence, there is \(f_k \in \nu _B\)  such that
\(f_k \circ \prod g_i \circ f_i (\sigma ^{\prime })\)  can be homotoped into \(\sigma ^{\prime }\)  and hence
\(f_k \circ \prod g_i \circ f_i \)  can be considered as a map with support in \(\sigma ^{\prime }\) .
Since, \(\sigma _n^{\prime }\)  has moved from \(A\)  to \(B\) , we in fact have \(f_k \circ \prod g_i \circ f_i= h_{\sigma ^{\prime }}^n\) .
That is \(h_{\sigma ^{\prime }}^n \in (\nu _L \mathcal {F})^{k+1}\) . This finishes the proof.
Proposition 7.3 
Assume \(\Sigma _i\)  is a surface with either zero or infinite genus and where \(E(\Sigma _i)\)  has a single maximal
point \(z\) . If \(h_\sigma \)  is essential, then \(E(z)\)  is two-sided giving the decomposition
\(\mathrm {Accu} (x) = X \sqcup Y\)  where \(x \in X\)  and \(y \in Y\) .

The proof is nearly identical to the proof of Proposition REF . We outline it here.
Let \(x\) , \(y\)  be the ends of \(\Sigma \)  associated to the strip \(\sigma \) . Let \(A, B \in \mathcal {A}\)  be
such that \(x \in A\)  and \(y \in B\) . First find \(g \in \operatorname{FMap} (\Sigma )\)  such that \(\sigma ^{\prime }=g(\sigma )\) 
starts in \(\Sigma _A\) , continues in \(L\) , and then enters \(\Sigma _B\) . It is enough to to show that
the group generated by \(h_{\sigma ^{\prime }} = g h_\sigma g^{-1}\)  is CB.
Note that \(x,y \in \mathrm {Accu} (z)\) . We can find a sequence
\(A = A_0, A_1, \dots , A_k = B, \qquad A_i \in \mathcal {A}\) 
and a sequence of ends \(z_i \in \mathrm {Accu} (z)\)  and \(z_i^{\prime } \in E(z_i)\) , \(i = 0, \dots , (k-1)\)  such that
\(z_0 \in A_0-\mathcal {M}(A_0)\) , \(z_{k-1}^{\prime } \in A_k-\mathcal {M}(A_k)\)  and, for \(i = 1, \dots , (k-1)\)  ,
\(z_{i-1}^{\prime }, z_i \in A_i-\mathcal {M}(A_{i-1})\) . A similar argument to the proof of Proposition REF 
shows that if such a sequence does not exist, then \(E(z)\)  is two-sided which would be a contradiction.
We then construct sub-surfaces \(Z_i\)  and \(Y_i\)  and maps \(g_i\)  as before, and set
\(\mathcal {F}= \lbrace  g_i\rbrace _{i=0}^{k-1}\) . Let \(\sigma _n^{\prime }\)  be a sub-strip of \(\sigma ^{\prime }\)  that contains
the \(n\)  sub-surfaces \(\Sigma _i\)  that are nearest to \(\partial \Sigma _A\) . As in the
proof of Proposition REF , we can choose \(f_i \in \nu _{A_i}\)  such that \(f_k \circ \prod (g_i \circ f_i)\) 
sends \(\sigma ^{\prime }\)  to the strip that is homotopic to \(\sigma ^{\prime }\)  and moves \(\sigma _n\)  from \(\Sigma _A\)  to
\(\Sigma _B\) . That is
\(h_{\sigma ^{\prime }}^n = f_k \circ \prod (g_i \circ f_i) \in \big (\nu _L \mathcal {F}\big )^{k+1}.\) 
This finishes the proof.
Proposition 7.4 
Assume \(\Sigma _i\)  is a surface with either zero or infinite genus, and where \(\mathcal {M}(E(\Sigma _i))\)  is a Cantor set.
Then \(h_\sigma \)  is not essential.

As before, after conjugation, we can assume \(\sigma \)  intersects only \(\Sigma _A \cup L \cup \Sigma _B\) , where
\(\Sigma _i \in \Sigma _A\)  for \(i \le 0\)  and \(\Sigma _i \in \Sigma _B\)  for \(i>0\) .
Consider a simple closed curve in the strip \(\sigma \)  separating \(\Sigma _i\)  and \(\Sigma _{i+1}\) 
from the rest of the strip. Denote this subsurface by \(\Sigma _{i, i+1}\) . If the set of maximal
ends in \(\Sigma _i\)  is a Cantor set, the end space of \(\Sigma _i\)  is homemorphic to the end
space of \(\Sigma _{i, i+1}\)  (see Figure REF ).
<FIGURE>Similarly, the subsurface \(\Sigma _{i, i+n}\) , which is a
subsurface
of \(\sigma \)  with one boundary component containing the sub-surfaces \(\Sigma _i, \Sigma _{i+1}, \dots ,\Sigma _{i+n}\) , is also homeomorphic to \(\Sigma _i\) . Hence, for every \(n\) , there is a homeomorphism
\(f_n \in \nu _A\)  with support in \(\sigma \)  such that \(\Sigma _i\)  is sent to \(\Sigma _{i+n}\)  for
\(i < -n\) , \(\Sigma _i\)  is sent to itself for \(i>0\) , and the surface \(\Sigma _{-n, 0}\)  is sent to
\(\Sigma _0\) . Similarly, there there is a homeomorphism
\(g_n \in \nu _A\)  with support in \(\sigma \)  such that \(\Sigma _i\)  is sent to \(\Sigma _{i+n}\)  for
\(i >1\) , \(\Sigma _i\)  is sent to itself for \(i\le 0\) , and the surface \(\Sigma _{1}\)  is sent to
\(\Sigma _{1,n+1}\) . Then \(h_\sigma ^n = g_n h_\sigma f_n\) . That is, we can collect the surfaces
\(\Sigma _{-n} \dots \Sigma _0\)  to the surface \(\Sigma _0\) , shift \(\sigma \)  once and the expand
these surfaces to \(\Sigma _1, \dots , \Sigma _{n+1}\) , which means we have written \(h_\sigma ^n\)  as a composition of 3
homeomorphisms. By letting \(\mathcal {F}= \lbrace  h_\sigma \rbrace \) , we have that
\(\langle h_\sigma \rangle \)  is contained in \((\nu _L \mathcal {F})^2\) . Therefore, by Theorem REF ,
\(\langle h_\sigma \rangle \)  is a CB subset of \(\operatorname{FMap} (\Sigma )\)  and \(h_\sigma \)  is not essential.
We are now in a position where we can prove Theorem REF , which states that \(\operatorname{Map} (\Sigma )\) 
contains an essential shift if and only if the endspace of \(\Sigma \)  is two-sided.
One direction is already proven by Proposition REF .
It remains to show that, if \(E\)  is not two-sided then there is no essential shift map.
Assume \(E^G\)  is not two-sided and there does not exist any \(z \in E\)  such that \(E(z)\)  is two-sided.
Let \(h_\sigma \)  be any shift map with support on a strip \(\sigma \) , containing the subsurfaces \(\Sigma _i\) ,
exiting towards \(x, y \in E\) . The end space \(E(\Sigma _i)\)  has finitely many maximal types each of which is either a
Cantor set or a finite set. Therefore we can find finitely many sub-surfaces \(\Sigma _i^j \subset \Sigma _i\) ,
\(j=0 , \dots , \ell \) , each with one boundary component such that:

The surfaces \(\Sigma _i^j\)  are disjoint. Furthermore, \(\Sigma - \Sigma _i^j\)  is a compact planar surface
which means \(E(\Sigma _i) = \sqcup _j E(\Sigma _i^j)\) .

For \(j>0\) , \(\Sigma _i^j\)  has zero genus or infinite genus. That is, if \(\Sigma _i\)  has finite genus
we include all the genus in the surface \(\Sigma _i^0\)  so every other subsurface of \(\Sigma _i\)  contains no genus. If
\(\Sigma _i\)  already has zero genus or infinite genus, then there is no need to choose \(\Sigma _i^0\) . We can choose
\(\Sigma _i^0\)  to be a disk or have \(j\)  range from 1 to \(\ell \) . Also, if \(E(\Sigma _i^j) \cap E^G =\emptyset \) 
then we make sure \(\Sigma _i^j\)  has genus zero.

For every \(j=0, \dots , \ell \) , the surfaces \(\Sigma _i^j\)  are homeomorphic for all \(i \in \mathbb {Z}\) .
That is we decompose each \(\Sigma _i\)  in the same way.

Now the strip \(\sigma \)  can be decomposed to parallel strips \(\sigma ^j\)  each containing sub-surfaces
\(\Sigma _i^j\) . Since \(h_{\sigma ^j}\)  have disjoint support, they commute and \(h_\sigma = \prod h_{\sigma ^j}\) .
That is, the group generated by \(h_{\sigma _j}\) , \(j=0, \dots , \ell \) , is an abelian group that contains
the group generated by \(h_\sigma \) . Hence, if each \(\langle h_{\sigma ^j} \rangle \)  is CB then
\(\langle h_{\sigma ^0, \dots , h_{\sigma ^\ell }} \rangle \)  is also CB and thus \(\langle h_{\sigma } \rangle \) 
is CB.
Since \(E^G\)  is not two-sided, by Proposition REF , \(h_{\sigma ^0}\)  is not essential and
\(\langle h_{\sigma ^0} \rangle \)  is CB. For \(j=1, \dots , \ell \) , if \(\mathcal {M}(E(\Sigma _i^j))\)  is a single
point then \(\langle h_{\sigma ^0} \rangle \)  is CB by Proposition REF . If \(\mathcal {M}(E(\Sigma _i^j))\) 
is a Cantor set, then \(\langle h_{\sigma ^0} \rangle \)  is CB by Proposition REF .
This finishes the proof.
Finally, we prove Theorem REF  which tells us that a shift map is essential
if and only if there is either a decomposition of the ends accumulated by genus, or there is a
decomposition of the accumulation set of a maximal point of the surface.
We begin by proving the forward direction.
Consider the decomposition of the shift map \(h_\sigma = \prod _j h_{\sigma ^j}\)  constructed in the
proof of Theorem REF . As mentioned before,
if \(h_\sigma \)  is essential then some \(h_{\sigma _j}\)  is essential. If \(h_{\sigma _0}\)  is essential, then since \(\Sigma _i^0\)  contains finite genus,
Proposition REF  implies the first bullet point in Theorem REF  holds. If \(j>0\)  and
\(\mathcal {M}(E(\Sigma _i^j))\)  is a single point, then by Proposition REF  the second bullet point in
Theorem REF  holds. Finally, \(\mathcal {M}(E(\Sigma _i^j))\)  cannot be a Cantor set
by Proposition REF , which proves the forward direction.
In the other direction, first suppose that \(E^G\)  is two-sided giving a decomposition \(E^G = X \sqcup Y\) 
where \(x \in X\)  and \(y \in Y\) , then the proof of Proposition REF 
shows that \(h_{\sigma ^0}\)  is essential. In fact, there is a length function \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)\(such that, for \) n>0\(, \) h0n = 2n\(. Since, in this case, \) ij\( have genus zerofor \) j>0\(, the shift maps \) hj\( act trivially on homology. Hence \) hn = 2n\(and hence \) h\( is essential.\) Similarly, if there exists some \(z \in \mathcal {M}(\Sigma _i)\)  such that \(E(z)\)  is two-sided giving a
decomposition \(\mathrm {Accu}(z)= X \sqcup Y\)  where \(x \in X\)  and \(y \in Y\) , then the proof of
Proposition REF 
shows that \(h_{\sigma ^j}\)  is essential for some \(j>0\) . In fact, there is a length function \({{\mathchoice{\hspace{0.55542pt}\mbox{$\centerdot $}}{\hspace{0.0pt}}{1}{m}u}{\hspace{0.55542pt}\mbox{$\centerdot $}}\hspace{0.55542pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}{\hspace{0.83328pt}\centerdot \hspace{0.83328pt}}\)\(such that, for \) n>0\(, \) h0n = n\(. It is possible that the \) ij\( are homeomorphicfor several different \) j\(. We can assume, after re-indexing, that \) i1, ..., i'\(all have same type of countable maximal end. Then \) hn = ' n\(and hence \) h\( is essential.\)
Two sources of non-trivial topology
In this section, we prove Theorem REF  which states that if \(\Sigma \)  does not
have a non-displaceable subsurface and \(\operatorname{FMap} (\Sigma )\)  does not have an essential shift then
\(\operatorname{Map} (\Sigma )\)  and \(\operatorname{FMap} (\Sigma )\)  are CB.
This implies that \(\operatorname{Map} (\Sigma )\)  and \(\operatorname{FMap} (\Sigma )\)  are quasi-isometric to a point and therefore do not have an interesting
geometry. To prove this theorem, we make use of the notion of an avenue surface first introduced in [3]}.
Recall that we always assume that \(\Sigma \)  is stable.
Definition 8.1 An avenue surface is a connected orientable surface \(\Sigma \)  which does not contain any
non-displaceable subsurface of finite type, and whose mapping class
group \(\operatorname{Map} (\Sigma )\)  is CB-generated but not CB.

That is, the only possible examples for surfaces that have no non-displaceable
subsurfaces, no essential shifts, and are not CB are the avenue surfaces.
A topological description of avenue surfaces was given in [3]}.
Lemma 8.2 (Lemmas 4.5 and 4.6 in [3]})  
Let \(\Sigma \)  be an avenue surface. Then \(\Sigma \)  has either 0 or infinite genus, and
\(\Sigma \)  has exactly two ends of maximal type, that is, \(\mathcal {M}(E)=\lbrace  x_1, x_2\rbrace \) .
Furthermore, for every \(x \in E-\lbrace x_1, x_2\rbrace \) , the set \(E(x)\)  accumulates to both \(x_1\) 
and \(x_2\) .

In order to state the classification of CB mapping class groups from [6]}, we must first
define the notion of self-similarity for a space of ends, and the notion of telescoping for
an infinite-type surface.
Definition 8.3 
A space of ends \((E, E^G)\)  is said to be self-similar if for any decomposition
\(E = E_1 \sqcup E_2 \sqcup \ldots \sqcup E_n\)  of \(E\)  into pairwise disjoint clopen sets,
there exists a clopen set \(D\)  in some \(E_i\)  such that \((D, D\cap E^G)\)  is homeomorphic to \((E, E^G)\) .
A surface \(\Sigma \)  is telescoping if there are ends \(x_1, x_2 \in E\)  and disjoint
clopen neighborhoods \(V_i\)  of \(x_i\)  in \(\Sigma \)  such that for all clopen neighborhoods
\(W_i \subset V_i\)  of \(x_i\) , there exist homeomorphisms \(f_i, h_i \in \operatorname{FMap} (\Sigma )\) , with
\(f_i(W_i) \supset (\Sigma - V_{3-i})\qquad h_i(W_i) = Vi,\qquad \text{and}\qquad h_i(V_{3-i}) = V_{3-i}.\) 

A classification of CB mapping class groups was given in [6]}.
Theorem 8.4 (Theorem 1.7 in [6]})  
The group \(\operatorname{Map} (\Sigma )\)  is CB if and only if \(\Sigma \)  has infinite or zero genus and \(E\) 
is either self-similar or telescoping.

We are now ready to prove Theorem REF  restated below.
Theorem REF . (Two sources of non-trivial geometry).

If \(\operatorname{Map} (\Sigma )\)  does not have an essential shift and \(\Sigma \)  does not contain
a non-displaceable subsurface then \(\operatorname{Map} (\Sigma )\)  is quasi-isometric to a point.
By way of contradiction, suppose that \(\Sigma \)  does not have a non-displaceable subsurface, \(\operatorname{FMap} (\Sigma )\) 
does not contain an essential shift, and that \(\operatorname{FMap} (\Sigma )\)  is not CB.
Then by assumption, \(\Sigma \)  is an avenue surface.
By Lemma REF ,
\(\mathcal {M}(E)=\lbrace x_1, x_2\rbrace \)  and for any other \(x \in E - \lbrace x_1, x_2\rbrace \) , \(E(z)\)  accumulates to
both \(x_1\)  and \(x_2\) . Consider an end \(z \in \mathcal {M}(E-\lbrace x_1, x_2\rbrace )\) , that is, an end that is
maximal in \(E-\lbrace x_1, x_2\rbrace \) . If \(E(z)\)  is countable, then \(E\)  is two-sided with
\(X=\lbrace x_1\rbrace \)  and \(Y=\lbrace x_2\rbrace \) . This cannot happen since we are
assuming there are no essential shifts (see Theorem REF ).
Therefore, \(E(z)\)  is a Cantor set.
Furthermore, either \(\Sigma \)  has genus zero or it is infinite genus
and \(\lbrace x_1, x_2 \rbrace  \subset E^G\)  (otherwise, \(\operatorname{Map} (\Sigma )\)  is not even locally CB, see
[6]}). We now show that these assumptions imply
that \(\Sigma \)  is telescoping and, by Theorem REF , \(\operatorname{Map} (\Sigma )\)  is CB, which will prove our claim.
We check the definition of telescoping. For \(i=1,2\) , let \(V_i\)  be disjoint stable neighborhoods
of \(x_i\)  such that \(E-(V_1 \cup V_2)\)  contains every maximal type in \(E-\lbrace x_1, x_2\rbrace \) .
Let \(W_i\)  be the given smaller neighborhoods of \(x_i\) . Since the \(V_i\)  are stable neighborhoods,
we have that \(V_i\)  is homeomorphic to \(W_i\) . We claim that \(E-(V_1 \cup V_2)\)  is homeomorphic to
\(E-(V_1 \cup W_2)\) . Since all maximal types in \(E-\lbrace x_1, x_2\rbrace \)  are present in \(E-(V_1 \cup V_2)\) ,
the sets \(E-(V_1 \cup V_2)\)  and \(E-(V_1 \cup W_2)\)  have the same maximal types. But all
these types are Cantor sets. Therefore, for every \(y \in V_2-W_2\)  there is a \(z \in E-(V_1 \cup V_2)\) 
such that \(E(y)\)  accumulates to \(z\) . By [6]}, there are small neighborhoods \(U_y\) 
of \(y\)  and \(U_z\)  of \(z\)  such that \(U_y \cup U_z\)  is homeomorphic to \(U_z\) . The neighborhoods
\(U_y\)  give a covering of \(V_2-W_2\) , hence there is a finite sub-covering. By making the neighborhoods
smaller, we can assume that they are disjoint. Since each neighborhood \(U_y\)  can be absorbed into \(U_z\) ,
the set \(V_2-W_2\)  can be absorbed into \(E-(V_1 \cup V_2)\)  and hence, \(E-(V_1 \cup V_2)\)  is
homeomorphic to \(E-(V_1 \cup W_2)\) .
Now consider surfaces \(\Sigma _{V_i}\)  and \(\Sigma _{W_i}\) 
whose end points are \(V_i\)  and \(W_i\) . We can choose these surfaces so that \(\Sigma _{W_i} \subset \Sigma _{V_i}\) 
and so that \(\Sigma _{V_i}\)  is disjoint from \(\Sigma _{V_{3-i}}\) . Also, we can assume
\(\Sigma _{V_i} - \Sigma _{W_i}\)  and \(\Sigma -(\Sigma _{W_1} \cup \Sigma _{W_2})\)  all have
genus zero or infinity. The end space of \(\Sigma _{V_i}- \Sigma _{W_i}\) 
is homeomorphic to the end space of \(\Sigma -(\Sigma _{W_1} \cup \Sigma _{W_2})\) 
and they have the same
genus. Also \(\Sigma _{V_1}\)  and \(\Sigma _{W_1}\)  have homeomorphic end spaces and the same
genus. Hence, there is a map \(h_1\)  such that \(h_1(\Sigma _{W_1})=\Sigma _{V_1}\)  and
\(h_1(\Sigma _{V_2}) = h_1(\Sigma _{V_2})\) . The map \(h_2\)  in Definition REF  can
also be similarly constructed.
The construction of the maps \(f_i\)  follows similarly. As above, we can send
\(\Sigma _{W_1}\)  to \(\Sigma -\Sigma _{V_2}\)  while fixing some neighborhood
\(W_2^{\prime }\)  of \(x_2\)  as long as \(W_2^{\prime }\)  is small enough so that \(V_2-W_2^{\prime }\)  intersect \(E(z)\)  for every
\(z \in \mathcal {M}(E-\lbrace x_1, x_2\rbrace )\) . This proves that \(\Sigma \) 
is telescoping and hence \(\operatorname{Map} (\Sigma )\)  is CB. Note that since
\(\operatorname{FMap} (\Sigma )\)  and \(\operatorname{Map} (\Sigma )\)  are quasi-isometric this means
\(\operatorname{FMap} (\Sigma )\)  is also CB which proves the theorem.
